Date: December 24 - 25

Venue: E4-233, Yungu Campus

ZOOM ID: 937 0545 8156

Passcode: 288289

Ⅰ  Agenda

Ⅱ  Talks

1. 8:30-10:00, December 24, Tuesday

Speaker: Hideo Kozono, Waseda University

Title: Asymptotic behavior of solutions to elliptic equations in 2D exterior domains

Abstract: The asymptotic behavior of solutions to the second order elliptic equations in exterior domains is studied.

In particular, under the assumption that the solution belongs to the Lorentz space or the weak Lebesgue space with certain conditions on the coefficients, we give natural and an almost sharp pointwise estimate of the solution at spacial infinity. The proof is based on the argument by Korobkov-Pileckas-Russo in which the decay property of the solution to the vorticity equation of the two-dimensional Navier-Stokes equations was studied.

This is based on the joint work with Yutaka Terasawa(Nagoya Univ.) and Yuta Wakasugi(Hiroshima Univ.)

2. 10:00-11:00, December 24, Tuesday

Speaker: Xiao Ren, Peking University

Title: Global solutions to the Euler-Coriolis system (Lecture Series)

Abstract: I will present a recent result on the global well-posedness and scattering of the Euler equations with a Coriolis force. Equivalently, we prove the global stability for the uniform rotating solution to the Euler equations. This extends the remarkable work of Guo, Pausader and Widmayer in the axisymmetric case to the general non-axisymmetric setting. The proof relies on an inherent null structure of the system, a delicate analysis of the spacetime resonances, and a multiscale wave-packet decomposition. Joint work with Prof. Gang Tian.

3. 11:00-12:00, December 24, Tuesday

Speaker: Yuanyuan Feng, East China Normal University

Title: Speeding up Langevin Dynamics by Mixing

Abstract: In this talk, we would study how stirring would help dissipate the energy and the applications, especially the application to the Langevin dynamics. We would first introduce the dissipation enhancing flows and study the dissipation time of such flows, where the dissipation time is explicitly computed based on the mixing rate. We then use mixing velocity field as a drift added to the Langevin dynamics (without changing the stationary distribution) and obtain quantitative estimates on the convergence rate of the system. We show that an exponentially mixing drift can be rescaled to make the mixing time of the Langevin system arbitrarily small.

4. 13:30-14:30, December 24, Tuesday

Speaker: Daoguo Zhou, Hangzhou Nornal University

Title: On regularity of the Navier-Stokes equations

Abstract: First, we establish local regularity criteria of "all scale" for suitable weak solutions to the 3D Naiver-Stokes equations, which does not involve the pressure. The proof is based on the local pressure projection due to Wolf and  energy related inequalities without pressure. Then, we prove logarithmic improvement of the classical Ladyzhenskaya-Prodi-Serrin criteria of Naiver-Stokes equations.

5. 14:30-15:30, December 24, Tuesday

Speaker: Hao Liu, Shanghai Jiao Tong University

Title: On the rigidity and existence of self-similar type solutions to the steady Navier-Stokes Equations

Abstract: Solutions with scaling-invariant bounds such as self-similar solutions, play important roles in the understanding of the regularity and asymptotic structures of solutions to the Navier-Stokes equations. In this talk, we will discuss some recent rigidity results about the solutions satisfying some scaling-invariant bounds to the steady Navier-Stokes equations. Then we talk about some applications of these rigidity results such as the regularity and the leading order behaviors of the solutions in the exterior domain. We may also talk about existence of large self-similar solutions to steady Navier-Stokes Equations with large external forces. This talk is based on the joint work with Jeaheang Bang, Changfeng Gui, Yun Wang and Chunjing Xie.

5. 15:30-16:30, December 24, Tuesday

Speaker: Deng Zhang, Shanghai Jiao Tong University

Title: Recent progresses on the stochastic Zakharov system

Abstract: In this talk we review very recent progresses on stochastic Zakharov systems in dimensions three and four. The Zakharov system couples Schroedinger and wave equations, and reaches the energy criticality in dimension four. We will mainly show the global well-posedness below the ground state and the noise regularization effects on blow-up and scattering dynamics.

6. 10:00-11:00, December 25, Wednesday

Speaker: Jeaheang Bang, Westlake University

Title: Self-Similar Solutions to the Stationary Navier-Stokes Equations in a Two-Dimensional Sector

Abstract: For the stationary Navier-Stokes equations, we investigated self-similar solutions in a two-dimensional sector with the no slip boundary condition. We found necessary and sufficient conditions for the existence of a self-similar solution in terms of the angle of the sector and the flux. In addition, we established the uniqueness and non-uniqueness of flows with a given type. As an application, we identified the leading order term of a solution to the stationary Navier-Stokes equations in an aperture domain when the flux is small. The main idea is to study the reduced ODE system and use properties of both complete and incomplete elliptic functions. This is a joint work with Changfeng Gui, Chunjing Xie, Yun Wang, and Hao Liu.

7. 11:00-12:00, December 25, Wednesday

Speaker: Zirong Zeng, Nanjing University of Aeronautics and Astronautics

Title: Existence and non-uniqueness of weak solutions with continuous energy to the 3D deterministic and stochastic Navier-Stokes equations

Abstract: In this talk, I will present our recently result about continuous energy solutions to the 3D deterministic and stochastic Navier-Stokes equations. The continuity of the kinetic energy is an important property of incompressible viscous fluid flows. We show that for any prescribed finite energy divergence-free initial data there exist infinitely many global in time weak solutions with smooth energy profiles to both the 3D deterministic and stochastic incompressible Navier-Stokes equations. In the stochastic case the constructed solutions are probabilistically strong. Our proof introduces a new backward convex integration scheme with delicate selections of initial relaxed solutions, backward time intervals, and energy profiles. Our initial relaxed solutions satisfy a new time-dependent frequency truncated NSE, different from the usual approximations as it decreases the large Reynolds error near the initial time, which plays a key role in the construction. This talk is based on the joint work with Alexey Cheskidov and Deng Zhang.